1 static char help[] = "Solves the van der Pol DAE.\n\ 2 Input parameters include:\n"; 3 4 /* ------------------------------------------------------------------------ 5 6 This program solves the van der Pol DAE 7 y' = -z = f(y,z) (1) 8 0 = y-(z^3/3 - z) = g(y,z) 9 on the domain 0 <= x <= 1, with the boundary conditions 10 y(0) = -2, y'(0) = -2.355301397608119909925287735864250951918 11 This is a nonlinear equation. 12 13 Notes: 14 This code demonstrates the TS solver interface with the Van der Pol DAE, 15 namely it is the case when there is no RHS (meaning the RHS == 0), and the 16 equations are converted to two variants of linear problems, u_t = f(u,t), 17 namely turning (1) into a vector equation in terms of u, 18 19 [ y' + z ] = [ 0 ] 20 [ (z^3/3 - z) - y ] [ 0 ] 21 22 which then we can write as a vector equation 23 24 [ u_1' + u_2 ] = [ 0 ] (2) 25 [ (u_2^3/3 - u_2) - u_1 ] [ 0 ] 26 27 which is now in the desired form of u_t = f(u,t). As this is a DAE, and 28 there is no u_2', there is no need for a split, 29 30 so 31 32 [ F(u',u,t) ] = [ u_1' ] + [ u_2 ] 33 [ 0 ] [ (u_2^3/3 - u_2) - u_1 ] 34 35 Using the definition of the Jacobian of F (from the PETSc user manual), 36 in the equation F(u',u,t) = G(u,t), 37 38 dF dF 39 J(F) = a * -- - -- 40 du' du 41 42 where d is the partial derivative. In this example, 43 44 dF [ 1 ; 0 ] 45 -- = [ ] 46 du' [ 0 ; 0 ] 47 48 dF [ 0 ; 1 ] 49 -- = [ ] 50 du [ -1 ; 1 - u_2^2 ] 51 52 Hence, 53 54 [ a ; -1 ] 55 J(F) = [ ] 56 [ 1 ; u_2^2 - 1 ] 57 58 ------------------------------------------------------------------------- */ 59 60 #include <petscts.h> 61 62 typedef struct _n_User *User; 63 struct _n_User { 64 PetscReal next_output; 65 }; 66 67 /* 68 User-defined routines 69 */ 70 71 static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx) 72 { 73 PetscScalar *f; 74 const PetscScalar *x, *xdot; 75 76 PetscFunctionBeginUser; 77 PetscCall(VecGetArrayRead(X, &x)); 78 PetscCall(VecGetArrayRead(Xdot, &xdot)); 79 PetscCall(VecGetArray(F, &f)); 80 f[0] = xdot[0] + x[1]; 81 f[1] = (x[1] * x[1] * x[1] / 3.0 - x[1]) - x[0]; 82 PetscCall(VecRestoreArrayRead(X, &x)); 83 PetscCall(VecRestoreArrayRead(Xdot, &xdot)); 84 PetscCall(VecRestoreArray(F, &f)); 85 PetscFunctionReturn(PETSC_SUCCESS); 86 } 87 88 static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx) 89 { 90 PetscInt rowcol[] = {0, 1}; 91 PetscScalar J[2][2]; 92 const PetscScalar *x; 93 94 PetscFunctionBeginUser; 95 PetscCall(VecGetArrayRead(X, &x)); 96 J[0][0] = a; 97 J[0][1] = -1.; 98 J[1][0] = 1.; 99 J[1][1] = -1. + x[1] * x[1]; 100 PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES)); 101 PetscCall(VecRestoreArrayRead(X, &x)); 102 103 PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)); 104 PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)); 105 if (A != B) { 106 PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)); 107 PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)); 108 } 109 PetscFunctionReturn(PETSC_SUCCESS); 110 } 111 112 static PetscErrorCode RegisterMyARK2(void) 113 { 114 PetscFunctionBeginUser; 115 { 116 const PetscReal A[3][3] = 117 { 118 {0, 0, 0}, 119 {0.41421356237309504880, 0, 0}, 120 {0.75, 0.25, 0} 121 }, 122 At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL; 123 PetscCall(TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL)); 124 } 125 PetscFunctionReturn(PETSC_SUCCESS); 126 } 127 128 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 129 static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, void *ctx) 130 { 131 const PetscScalar *x; 132 PetscReal tfinal, dt; 133 User user = (User)ctx; 134 Vec interpolatedX; 135 136 PetscFunctionBeginUser; 137 PetscCall(TSGetTimeStep(ts, &dt)); 138 PetscCall(TSGetMaxTime(ts, &tfinal)); 139 140 while (user->next_output <= t && user->next_output <= tfinal) { 141 PetscCall(VecDuplicate(X, &interpolatedX)); 142 PetscCall(TSInterpolate(ts, user->next_output, interpolatedX)); 143 PetscCall(VecGetArrayRead(interpolatedX, &x)); 144 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[%.1f] %3" PetscInt_FMT " TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n", (double)user->next_output, step, (double)t, (double)dt, (double)PetscRealPart(x[0]), (double)PetscRealPart(x[1]))); 145 PetscCall(VecRestoreArrayRead(interpolatedX, &x)); 146 PetscCall(VecDestroy(&interpolatedX)); 147 user->next_output += PetscRealConstant(0.1); 148 } 149 PetscFunctionReturn(PETSC_SUCCESS); 150 } 151 152 int main(int argc, char **argv) 153 { 154 TS ts; /* nonlinear solver */ 155 Vec x; /* solution, residual vectors */ 156 Mat A; /* Jacobian matrix */ 157 PetscInt steps; 158 PetscReal ftime = 0.5; 159 PetscBool monitor = PETSC_FALSE; 160 PetscScalar *x_ptr; 161 PetscMPIInt size; 162 struct _n_User user; 163 164 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 165 Initialize program 166 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 167 PetscFunctionBeginUser; 168 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 169 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); 170 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!"); 171 172 PetscCall(RegisterMyARK2()); 173 174 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 175 Set runtime options 176 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 177 178 user.next_output = 0.0; 179 PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL)); 180 181 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 182 Create necessary matrix and vectors, solve same ODE on every process 183 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 184 PetscCall(MatCreate(PETSC_COMM_WORLD, &A)); 185 PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2)); 186 PetscCall(MatSetFromOptions(A)); 187 PetscCall(MatSetUp(A)); 188 PetscCall(MatCreateVecs(A, &x, NULL)); 189 190 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 191 Create timestepping solver context 192 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 193 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); 194 PetscCall(TSSetType(ts, TSBEULER)); 195 PetscCall(TSSetIFunction(ts, NULL, IFunction, &user)); 196 PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user)); 197 PetscCall(TSSetMaxTime(ts, ftime)); 198 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER)); 199 if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL)); 200 201 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 202 Set initial conditions 203 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 204 PetscCall(VecGetArray(x, &x_ptr)); 205 x_ptr[0] = -2; 206 x_ptr[1] = -2.355301397608119909925287735864250951918; 207 PetscCall(VecRestoreArray(x, &x_ptr)); 208 PetscCall(TSSetTimeStep(ts, .001)); 209 210 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 211 Set runtime options 212 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 213 PetscCall(TSSetFromOptions(ts)); 214 215 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 216 Solve nonlinear system 217 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 218 PetscCall(TSSolve(ts, x)); 219 PetscCall(TSGetSolveTime(ts, &ftime)); 220 PetscCall(TSGetStepNumber(ts, &steps)); 221 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "steps %3" PetscInt_FMT ", ftime %g\n", steps, (double)ftime)); 222 PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD)); 223 224 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 225 Free work space. All PETSc objects should be destroyed when they 226 are no longer needed. 227 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 228 PetscCall(MatDestroy(&A)); 229 PetscCall(VecDestroy(&x)); 230 PetscCall(TSDestroy(&ts)); 231 232 PetscCall(PetscFinalize()); 233 return 0; 234 } 235 236 /*TEST 237 238 test: 239 requires: !single 240 suffix: a 241 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp 242 output_file: output/ex19_pi42.out 243 244 test: 245 requires: !single 246 suffix: b 247 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42 248 output_file: output/ex19_pi42.out 249 250 test: 251 requires: !single 252 suffix: c 253 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_pid 0.4,0.2 254 output_file: output/ex19_pi42.out 255 256 test: 257 requires: !single 258 suffix: bdf_reject 259 args: -ts_type bdf -ts_dt 0.5 -ts_max_steps 1 -ts_max_reject {{0 1 2}separate_output} -ts_error_if_step_fails false -ts_adapt_monitor 260 261 TEST*/ 262